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%I #29 Feb 07 2021 14:14:45
%S 1,1,4,2,48,16,576,144,3840,768,9216,1536,3870720,552960,442368,55296,
%T 26542080,2949120,2229534720,222953472,70071091200,6370099200,
%U 76441190400,6370099200,16694755983360,1284211998720,570760888320
%N The n-th Minkowski number divided by the n-th factorial: a(n) = A053657(n)/n!.
%C a(n) is an integer by Legendre's formula for the exponent of the highest power of a prime dividing n!.
%C a(2n-1) = n*a(2n) because A053657(2n) = 2*A053657(2n-1).
%C See A053657 for additional comments, references, and links.
%H F. Bencherif, <a href="http://at.yorku.ca/c/a/y/d/47.htm">Sur une propriété des polynômes de Stirling</a>, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France.
%H Farid Bencherif, Tarek Garici, <a href="https://doi.org/10.2298/PIM1716149B">On a property of Stirling polynomials</a>, Publications de l'Institut Mathématique (2017), Vol. 102, Issue 116, pp. 149-153.
%H J.-L. Chabert, <a href="http://dx.doi.org/10.1016/j.ejc.2005.12.009">Integer-valued polynomials on prime numbers and logarithm power expansion</a>, European J. Combinatorics 28 (2007) 754-761.
%F a(n) = A053657(n)/A000142(n).
%F a(n) = (1/n!)*Product_{p prime} p^(Sum_{k>=0} ((n-1)/((p-1)p^k))).
%F a(n) = (1/n!)*denominator([y^(n-1)](y/(exp(y)-1))^x). - _Peter Luschny_, May 13 2019
%e a(4) = A053657(4)/4! = 48/24 = 2.
%p A163176 := proc(n) local L,p;
%p L := proc(n,p,r) local q,s; q := p-r; s := 0;
%p do if q > n then break fi; s := s+iquo(n,q);
%p q := q*p od; s end; mul(p^(L(n-1,p,1)-L(n,p,0)),
%p p = select(isprime,[$2..n])) end: # _Peter Luschny_, Jul 26 2009
%p ser := series((y/(exp(y)-1))^x, y, 29):
%p c := n -> denom(coeff(ser, y, n-1)):
%p seq(c(n)/n!, n=1..27); # _Peter Luschny_, May 13 2019
%t a[n_] := (1/n!)*Product[ p^Sum[ Floor[ (n-1)/((p-1)*p^k) ], {k, 0, n}], {p, Select[ Range[2, n], PrimeQ]}]; Table[ a[n], {n, 1, 27}] (* _Jean-François Alcover_, Dec 07 2011 *)
%o (Julia)
%o using Primes
%o function A163176(n)
%o function L(n, p, r)
%o s, q = 0, p - r
%o while q <= n
%o s += div(n, q)
%o q *= p
%o end
%o s end
%o n < 2 && return 1
%o P = primes(n)
%o prod(p^(L(n-1, p, 1) - L(n, p, 0)) for p in P)
%o end
%o [A163176(n) for n in 1:27] |> println # _Peter Luschny_, Feb 07 2021
%Y Cf. A000142 (n!), A053657.
%K easy,nonn
%O 1,3
%A _Jonathan Sondow_, Jul 24 2009
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